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Regularity conditions and the maximum likelihood estimation in dynamical systems with small fractional Brownian noise. (English) Zbl 1115.62080

The author deals with a dynamical system with small fractional Brownian noise, i.e., a process described by the equation \[ X_t=x_0+\int_0^tS(\theta,u,X_u)\,du+\varepsilon B_t,\quad 0\leq t\leq T, \] where \(B_t\) is a fractional Brownian motion (fBm) with the Hurst parameter \(H\in(1/2,1)\) and \(\theta\in\Theta\subset\mathbb R^d\) is an unknown parameter. Under some, not too restrictive, conditions on the function \(S\) there exists a unique solution \(X\) to the equation in a pathwise sense for every \(\theta\in\Theta\subset\mathbb R^d\) and \(\varepsilon>0\). Following the terminology of I. A. Ibragimov and R. Z. Khas’minskij [Asymptotic Theory of Estimation. (Russian) (1979; Zbl 0467.62025)] we may say that the equation generates a set of statistical experiments \[ \mathbb E_{\varepsilon}=\left\{C([0,T]),\,\mathcal B,\,P_{\theta}^{(\varepsilon)},\;\theta\in\Theta \right\}, \] in which we observe the trajectory of a solution \(X=X^{\varepsilon}\). The index \(\varepsilon>0\) refers to the noise intensity in the experiment. The author presented sufficient conditions under which this dynamical system with small fractional Brownian noise generates a set of regular statistical experiments in the sense of I. A. Ibragimov and R. Z. Khas’minskij definition. As a corollary, it is shown that the maximum likelihood estimator of the unknown parameter based on the observation of a trajectory is consistent, uniformly asymptotically normal and its moments converge to moments of the standard normal distribution.

MSC:

62M05 Markov processes: estimation; hidden Markov models
62M09 Non-Markovian processes: estimation
62F12 Asymptotic properties of parametric estimators
60G15 Gaussian processes
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)

Citations:

Zbl 0467.62025

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References:

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